There are certain things in mathematics that have caused me a pleasant surprise -- when some part of mathematics is brought to bear in a fundamental way on another, where the connection between the two is unexpected. The first example that comes to my mind is the proof by Furstenberg and Katznelson of Szemeredi's theorem on the existence of arbitrarily long arithmetic progressions in a set of integers which has positive upper Banach density, but using ergodic theory. Of course in the years since then, this idea has now become enshrined and may no longer be viewed as surprising, but it certainly was when it was first devised.

Another unexpected connection was when Kolmogorov used Shannon's notion of probabilistic entropy as an important invariant in dynamical systems.

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As well known as the connection is, I am constantly amazed by the power of analytical geometry (developed by Descartes and Fermat) to make connections between geometrical ideas and algebraic ideas. It seems remarkable to me that so much geometrical information (as for example in the case of the conic sections) can be represented so succinctly (via quadratic equations in two variables). The geometry suggests things to think about in algebra and the algebra suggests things to think about in geometry. It is just amazing!!

This is the observation that should have occurred to everyone first! (It didn't to me either.) It is so familiar we forget how amazing it is.
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SixWingedSeraphFeb 8 '10 at 3:02

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I totally agree. To put another exclamation point on this idea, Algebra and Geometry co-existed for around a thousand years before this observation was made mainstream by Descartes and Fermat. I wonder what other yet-unseen mathematics we'll weave, in a thousand years, into middle school education.
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Hiro Lee TanakaSep 29 '13 at 22:34

I mean why should the Fourier series of the $j$-invariant have coefficients related to the dimensions of the representations of the largest sporadic simple group? And why should the proof of this fact drag in mathematics from String Theory?

I'll recycle one I mentioned in a thread last week, connecting an elementary problem about polynomials to the classification of finite simple groups:

Definition: A polynomial $f(x) \in \mathbb{C}[x]$ is indecomposable if whenever $f(x) = g(h(x))$ for polynomials $g$, $h$, one of $g$ or $h$ is linear.

Theorem. Let $f, g$, be nonconstant indecomposable polynomials over $\mathbb C$. Suppose that $f(x)-g(y)$ factors in $\mathbb{C}[x,y]$. Then either $g(x) = f(ax+b)$ for some $a,b \in \mathbb{C}$, or
$$\operatorname{deg} f = \operatorname{deg} g = 7, 11, 13, 15, 21, \text{ or } 31,$$
and each of these possibilities does occur.

The proof uses the classification of the finite simple groups [!!!] and is due to Fried [1980, in the proceedings of the 1979 Santa Cruz conference on finite groups], following a the reduction of the problem to a group/Galois-theoretic statement by Cassels [1970]. [W. Feit, "Some consequences of the classification of finite simple groups," 1980.]

My favorite surprise, which is perhaps the record-holder for the longest time
it took for the two ideas to be brought together, is the connection between
regular n-gons and Fermat primes. The Greeks knew how to construct regular
n-gons by ruler and compass for n=3,4,5,6. Fermat introduced numbers of the
form $2^{2^m}+1$ around 1640 in the mistaken belief they were prime for all m.
Then in 1796 Gauss discovered how to construct the regular 17-gon, and a few
years later showed that the n in a constructible n-gon is the product of some
power of 2 by distinct Fermat primes.

Jacobi noted, as mathematics' most fascinating property, that in it one and the same function controls both the presentations of a whole number as a sum of four squares and the real movement of a pendulum.

Quillen's result that the ring of cobordism classes of (stably) complex manifolds is isomorphic to Lazard's ring (i.e. the universal ring classifying formal group laws). This seems so mysterious to me. Why should cobordism classes of complex manifolds have anything to do with the algebraic geometry of formal group laws? Nevertheless this has been one of the most important observations for modern homotopy theory. It is the driving force behind Chromatic Stable Homotopy which tries to build a dictionary between the algebraic geometry of FGLs and structures present in the stable homotopy category. It is shocking how successful this has been.

Here is one of my favorites. If you consider a singular node of an algebraic curve locally it looks like the curves $xy=0$ in $\mathbb{C}^2$, or $x^2+y^2=0$. This consists of two smooth arcs intersecting to each other transversally (reducible in particular).

Now, one step further, if we consider a cusp which is analytically equivalent to the origin in the curve $y^2+x^3=0$ in $\mathbb{C}^2$, it is locally irreducible. However, here comes the interesting point, if we intersect the singularity with a small ball $$[(x,y)\in \mathbb C^2:\ |x|^2+|y|^2=\epsilon]\cong S^3$$ what we've got is that such an intersection is $$(ae^{2i\theta},a^{3/2}e^{3i\theta})\subset S^1\times S^1\subset S^3$$ which is contained in a torus winding two times in one direction in the torus and three times in the other direction, in other words, we have an trefoil knot.

Now in the case of surfaces, all these facts give rise to an amazing relation between topology and algebraic geometry. The underlaying space topological space in $\mathbb C^4$ of $$x^2+y^2+z^2+w^3=0$$ is a manifold!! (note it is singular at the origin in the context of AG!). As far as I know, if one intersects a small ball with the singularity, as I did above, one gets a topological sphere whose differential structure is NOT the standard one. Even more, considering in $\mathbb C^5$ the following hypersurface $$x^2+y^2+z^2+w^3+t^{6k-1}=0$$ and carrying out the intersection with a small sphere around the origin, for $k=1,2,\ldots 28$ one may get all the 28 possible exotic differential structures on the 7-sphere that Milnor found.

Wow,that IS pretty whack,Csar.This example alone is a testament to the power of modern topology and geometry and the incredible connections it has uncovered.
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Andrew LJul 15 '10 at 19:55

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Your claim about taking the link of the singular point of the hypersurface $x^2 + y^2 + z^2 + w^3 = 0$, and getting an exotic five-sphere, sounds wrong to me. Is there a reference for this?
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Sam NeadMar 13 '11 at 22:08

In my opinion, this becomes a lot less mysterious as soon as you think of the exponential and trigonometric functions as eigenfunctions of the differentiation operator (respectively, its square), which is really the reason they're both so important. The basic properties and interrelationships of these functions - including the above identity - are natural consequences of this formulation.
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Robin SaundersJul 29 '11 at 3:06

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Well yes, but that's the way it is with all these surprising results, isn't it? They all indicate a connection that no one had suspected, but is undoubtedly important. Once that connection is chewed over enough and becomes something you learn as a matter of course, then the original surprising result becomes "understandable", or sometimes even "trivial". But it certainly wasn't originally, and often isn't even to people first encountering these things today.
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Carl OffnerSep 30 '13 at 0:54

The ubiquity of Littlewood-Richardson coefficients. Given three partitions $\lambda, \mu, \nu$ each with at most $n$ parts, there is a combinatorial definition for a number $c^\nu_{\lambda, \mu}$ which is nonzero if and only if any of the following statements are true:

The irreducible representation of ${\bf GL}_n({\bf C})$ with highest weight $\nu$ is a subrepresentation of the tensor product of those irreducible representations with highest weights $\lambda$ and $\mu$.

Indexing the Schubert cells of the Grassmannian ${\bf Gr}(d,{\bf C}^m)$ (where $d \ge n$ and $m-d$ is at least as big as any part of $\lambda, \mu, \nu$) by $\sigma_\lambda$ appropriately, the cycle $\sigma_\nu$ appears in the intersection product $\sigma_\lambda \sigma_\mu$.

The beautiful analogy between number fields and function fields (and in general, algebra and geometry) that one learns about in arithmetic algebraic geometry.

Some specific examples:

The idea that a Galois group and a fundamental group (one algebro-number theoretic, the other geometric/topological) are two instances of the same thing.

The use of the term ramification in both number theory and geometry. Describing $\mathbb{Z}$ as simply connected because $\mathbb{Q}$ has no unramified extensions.

The appearance of integral closure in both algebraic geometry and algebraic number theory. The integral closure, in the former case, actually corresponds to a distinct geometric idea: non-singularity.

The idea of considering a prime number to be a point; then viewing localization at that prime, -adic completion at that prime, and the residue field of that prime as if they were the corresponding geometric objects. In particular, using the term "local" in number theory, as if we were talking about geometry! This idea is built into scheme theory.

There are many more examples.

This book looks deeply into the relationships between Galois groups and fundamental groups and eventually develops a theory which covers both.

This book explores the beautiful relation between algebraic curves and algebraic number theory.

I think the disparity between the world-views in low-dimensional topology versus high-dimensional topology are surprising. Even after you learn the reasons why, IMO they should still be surprising. Examples:

1) Teichmuller space exists, yet hyperbolic manifolds in dimension $3$ and larger are rigid. There are many interesting connections here such as the link between conformal geometry, complex analysis and hyperbolic geometry in dimension 2.

2) Exotic smooth structures on $\mathbb R^4$ but not on $\mathbb R^n$ for $n\neq 4$.

3) Why the Poincare conjecture/hypothesis is "hard" in dimensions $3$ and $4$ yet relatively "easy" in other dimensions.

4) Geometry being particularly relevant to $2$ and $3$-dimensional manifolds yet less so in higher dimensions.

I could go on. Some of these are connections, some I suppose are disconnections. But a connection is only a surprise if you have reason to think otherwise. :)

@Ryan: is the fact that geometry is less useful in high dimensions an empirical observation, or is there more mathematical content to this?
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Jim ConantMar 13 '11 at 17:44

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@Jim: From Hillman's "Four-manifolds, Geometries and Knots", up to homeomorphism there are only 11 geometric 4-dimensional manifolds with finite fundamental group. In dimension 4 a finite-volume hyperbolic manifold's volume is a function of its Euler characteristic. I see those as having a fair bit of content. Sorry for being slow to reply.
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Ryan BudneySep 1 '11 at 6:40

Connection between the typical number of isolated nonzero solutions ($N$) of a system of equations $$f_1=f_2=\cdots=f_n=0,$$
where each $f_k$ is a polynomial in $n$ complex variables,
and the mixed volume ($V$) of the Newton polytopes of $f_k$:
$$N=(n!){\cdot}V.$$

If you evaluate (appropriately normalized) elliptic functions at points lying in quadratic imaginary fields, the values you obtain are algebraic numbers, lying in abelian extensions of said quadratic imaginary fields; and all such extensions can be obtained in this way. (Compare with: $e^{2\pi i z}$ evaluated at rational numbers gives algebraic numbers, which generate abelian extension of ${\mathbb Q}$, and all abelian extension of ${\mathbb Q}$ are obtained in this way.
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EmertonFeb 8 '10 at 5:03

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Another way of saying it is that (with some slight fiddling) coordinates of points of finite order on an elliptic curve with complex multiplication give abelian extensions of the appropriate quadratic imaginary field. This was Kronecker's Jugendtraum (dream of his youth). Only in few cases is this explicit description of abelian extensions possible. Why do I think it is surprising? Compare what Emerton said above, $\exp(2 \pi iz)$ generating abelian extensions of $\mathbb{Q}$. Tell this to someone and ask them to guess how you'd generalise! It really is surprising that it is possible at all.
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Sam DerbyshireFeb 8 '10 at 8:37

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Elliptic curves (over $\mathbb{C}$) have their origins in studying elliptic integrals. As analytic objects they are $\mathbb{C}$ modulo a lattice. It's not immediately obvious to me that this is an algebraic object, and that the Weierstrass $\mathcal{P}$ function, which is an infinite sum, should compute anything number theoretic. So perhaps the connection is between analysis and algebra/arithmetic from this point of view.
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ZavoshFeb 8 '10 at 21:16

The work of Nabutovsky and Weinberger applying computability theory (a.k.a. recursion theory) to differential geometry. For example one of their results is that if you consider the space of Riemannian metrics on a smooth compact manifold $M$ of dimension at least 5 and sectional curvature $K\le 1$, then there are infinitely many extremal metrics. This is a purely geometric statement, but the only known proof uses concepts from computability theory. Moreover the results from computability theory that are used in their work are very deep; prior to their work, some skeptics regarded this area of computability theory as being overly specialized and having no hope of being connected to other areas of mathematics. See the exposition of Robert Soare (available on his website) for more information.

McKay's observation that the special fiber in the desingularization of du Val singularities is a bunch of $\mathbb P^1$s linked according to the Dynkin diagram corresponding to the group of the singularity.

It is possible to compute the Betti numbers of a smooth complex variety $X(\mathbb{C})$ by computing the cardinality of $X(\mathbb{F}_{p^n})$ for a prime $p$ with good reduction and a finite number of positive integers $n$; in other words, by brute force.

The above claim is wrong, so I'll phrase it the other way around. The Betti numbers of a smooth complex variety control the behavior of the number of points on $X(\mathbb{F}\_{p^n})$; for example, for a smooth projective curve of genus $g$ we have $|\text{Card}(X(\mathbb{F}\_q))| - q - 1| \le 2g \sqrt{q}$.

Generally I find the relationship between the arithmetic and topological properties of varieties surprising, although maybe this is a temporary kind of surprise that arithmetic geometers are used to. Another example: if $X$ is a curve, then whether the curvature of $X(\mathbb{C})$ is positive, zero, or negative determines whether $X(\mathbb{Q})$ is rationally parameterizable, a finitely generated group, or finite (unless it's empty).

This is probably not the most serious of applications, but I found the equivalence (in game theory) of the determinacy of Nash's board game Hex with the Brouwer Fixed Point theorem to be a surprising, if somewhat lighthearted, connection.

A caveat, this is not a theorem. Montgomery showed this for test functions whose Fourier transform had restricted support (in fact, support [-2,2] iirc.) Montgomery conjectured the same holds for more general test functions. Odlyzko's computations provided spectacular numerical evidence. And Katz-Sarnak proved an analogous statement for function fields.
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StoppleMar 13 '11 at 20:02

The fastest known solution of the word problem in braid groups originated from research on large cardinal axioms; the proof is independent of the existence of large cardinals, although the first version of the proof did use them. See Dehornoy, From large cardinals to braids via distributive algebra, Journal of knot theory and ramifications, 4, 1, 33-79.

To me this is an absolute mystery! Large cardinals are usually considered an esoteric subject situated on the border of the observable universe. So why should they have any relevance to braids, a very down to earth part of mathematics? Let alone give an algorithm for distinguishing braids, and what's more, the fastest algorithm known.

I'm not sure why this is surprising. It was originally defined via subfactors, but the path integral formalism followed very closely behind. Also, I'm not sure that Feynman path integrals count as mathematics...
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Daniel MoskovichFeb 8 '10 at 5:07

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Well, you are right. I think "surprising" is a subjective property, perhaps an experience of facing one's own ignorance. I don't know much about subfactors and first saw this polynomial in the context of knot invariants, divorced from its origins. For this reason, the physics connection seemed like a big surprise. It appears that you are an expert and so it is not surprising to me that you are not surprised.
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ZavoshFeb 8 '10 at 15:50

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I always found it ironic that knot theory began with Lord Kelvins model of atoms as knots in ether (loosely speaking). After a 360 degree rotation (or make that 720 degree :-) we're at string theory now.
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Hauke ReddmannJul 25 '11 at 12:00

Ulam's problem on determining the length of the longest increasing subsequence of a random permutation. The solution and the full description of the answer brought together ideas from integrable systems, combinatorics, representation theory, probability (appearing in the form of polynuclear growth model for instance), and random matrix theory.

Why down vote? I thought Jacques Hadamard expressed in his quote that in his time the connection of prime numbers to the zeroes of the Riemann zeta function was surprising and much of a shortcut to proving the Prime Number Theorem.
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UnknownJun 2 '10 at 1:05

Deligne's work was about counting solutions to equations over finite fields. Ramanujan's conjecture was about bounding the absolute values of the Fourier coefficients of a certain complex analytically defined function. How is the connection possibly tautological?
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EmertonFeb 8 '10 at 5:14

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Indeed, the connection between the Weil conjectures (and in particular the Riemann hypothesis, the proof of which is the work of Deligne being referred to) and Ramanujan's conjecture was only made some time after both conjectures were formulated (by Serre, I believe).
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EmertonFeb 8 '10 at 5:17

There exist two binary trees with rotation distance $2n-6$. The proof is unexpected and based on hyperbolic geometry (Sleator, Tarjan, Thurston (1988), "Rotation distance, triangulations, and hyperbolic geometry").

I was recently amazed at a quick connection between two facts I've known since high school. The Euler characteristic of a sphere, thought of as #vertices + #faces - #edges on a polyhedron, buckyball, etc., is 2; I re-deduced this from the fact that the derivative of $f(x)=1/x$ is $f'(x)=-1/x^2$.

The steps of the proof are as follows: construct the Riemann sphere using two complex charts, both C, with the holomorphic transition map $f(z)=1/z$ on each neighborhood minus its origin. Now we want to look at the Chern class of the cotangent bundle, which in standard orientation is the negative of the Euler class of the tangent bundle, i.e. the sphere. Well, assuming complex analysis, look at $df=\frac{-1}{z^2}dz$ to see the effect of the transition map on the cotangent bundles: as a ``holomorphic'' 1-form, that has a double pole at one point and no zeros. Thus we know that a section of the cotangent bundle of the sphere has divisor degree $-2$. So $\chi(S^2)=2$ and I now cannot separate this fact from $f'(x)=-1/x^2$ in my mind. It seem somehow more mysterious, ridiculous, and delightful that this connection is so short.

(Everyone I've mentioned this to prefers their own proof and perhaps it's better to do this slightly more directly to get a self-intersection 2 for a section of the tangent bundle, i.e. vector fields vanish twice, which gives the Euler class in $H^2(S^2)$ as a multiple of the orientation class.)

Goppa’s construction of error-correcting codes from curves, leading to the Tsfasman-Vladut-Zink bound (the first improvement over the Gilbert-Varshamov bound). An error-correcting code may be regarded as a combinatorial structure, and I think that this is a surprising connection between algebraic geometry and combinatorics.

To me, this isn't really shocking. It's a natural consequence of the cute (and, yes, maybe even surprising) fact that the square of $\int e^{x^2}\;dx$ is equal to $\int e^{x^2 + y^2}\;dx\;dy$, the integral of a function whose level sets are circles.
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VectornautFeb 1 '12 at 21:45